2
votes
0answers
29 views

Partition function proof

I am looking for any online information regarding Hardy and Ramanujan's proof, perhaps the proof itself, that the partition function $p(n)$ is asymptotic to $$\frac{e^{K\sqrt{n}}}{4n\sqrt{3}}$$ where ...
1
vote
1answer
30 views

Best self study book with answers to selected questions for analytic number theory?

All: Can anyone recommend Best self study book with answers to selected questions for analytic number theory ? If a book have no answers to questions, but if you know if some professors choose the ...
0
votes
1answer
24 views

which algebraic number theory book with answers to selected questions for self-study?

All: Can anyone recommend some easy to follow algebraic number theory books with answers (hints) to selected questions for self-study ? If a have no answers to questions, but if you know if some ...
13
votes
6answers
973 views

Is there any book/resource which explain the general idea of the proof of Fermat's last theorem?

I look for a book/resource which display the general idea of the proof of Fermat last theorem in a simple manner for the public. I mean, books which is not for mathematicians but for the general ...
1
vote
0answers
20 views

Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
1
vote
1answer
56 views

Periodicity with irrational numbers

Recently, I invented the following theorem and found a proof, it seems strange since it is very counter-intuitive to me. The proof is long and non-conceptual. Is there a place or a branch of math ...
0
votes
1answer
81 views

Who should I ask for Robin's paper? At any rate, I want to find out if a similar result to his can be achieved with 36 instead of 12.

Robin proved unconditionally that for $\ n \ge 3$ , $$ \sigma(n)<\left(e^\gamma+{\log\log12\left({\frac73}-e^\gamma \log\log12\right)\over (\log \log n)^2}\right)n \log \log n. $$ I need a similar ...
1
vote
0answers
72 views

Any Computational Number Theory Book, include software programs for key steps of the proofs of major theorem?

All: Can anyone recommend some Computational Number Theory Books, which include software programs for key steps of the proofs of major theorem ? Some computational number theory books only include ...
0
votes
0answers
21 views

What is the latest that we know about odd perfect numbers?

What is the latest that we know about odd perfect numbers? I have seen some recent papers by Nielsen, and Ochem & Rao. I was wondering if there are any others.
1
vote
1answer
50 views

On some inequalities which are an generalization of F. Beukers' corresponding results

In 1981, F. Beukers proved the following theorem in his article On the generalized Ramanujan-Nagell equation I : Theorem $\bf1$. $~$ Suppose $m \in \mathbb{Z}$, then for any integer $x$, $$ ...
1
vote
0answers
19 views

Bounds re Asymptotic Formula for the Sum of Largest Prime Factors

I have a reference request related to the result : $\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$ where $P(n)$ is the largest prime factor of the positive ...
4
votes
2answers
80 views

what is the most easy to read Algebraic Geometry book? [duplicate]

All: what is the most easy to read (most accessible) Algebraic Geometry book ? (If possible, I am looking for an introduction book, maybe for undergraduate, and maybe similar to A Friendly ...
2
votes
0answers
46 views

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer

As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,1,6,1,2,1,6…]$,let $L$ the length of period of simple continued fraction expansion of quadratic algebraic numbers be the number of ...
0
votes
0answers
93 views

Proof that $G(3)\le 7$

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers. Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...
0
votes
0answers
51 views

Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...
3
votes
0answers
34 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
0
votes
1answer
71 views

Can anyone recommend an easy to read algebraic number theory book?

Can anyone recommend an easy to read algebraic number theory book ? I prefer a book with good examples. (hints or answers to selected questions if possible. Not sure if it is possible for a book of ...
1
vote
2answers
119 views

Property of set of prime numbers

let $\{p_1,p_2,p_2,\cdots ,p_r\}$ be the set of $r$($\ge2$) pair wise distinct prime numbers i.e.., $(i\ne j \implies p_i \ne p_j)$ for all $1\le i,j\le r$ ${Statement}$ : For any such ...
1
vote
0answers
34 views

Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
1
vote
0answers
19 views

What is the definition of period number and any relation between Abelian integral and such a kind of period number

I recall there is a kind of real or complex number called period number which is defined by integral and algebraic function.But now,I search it again and again having gotten no result.Now any one can ...
6
votes
1answer
155 views

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
1
vote
1answer
44 views

Generalization for Stirling numbers 2nd kind to negative column-indexes?

The exponential generating functions for the Stirling numbers 2nd kind are the n'th powers of $f(x)=\exp(x)-1$ (where this is understood as formal power series, Abramowitz&Stegun, 26.8.12). ...
1
vote
2answers
42 views

Compendium of elliptic curves?

does anyone know where I can find a collection of elliptic curves and their integral solutions? EDIT: Removed additional useless info. Thanks!
1
vote
1answer
65 views

Is $\{1,1,2,3,4,5,\cdots,i,\cdots \} $ the simple continued fraction algebraic or transcendental?

Is $$1+\cfrac{1}{1+\cfrac{1}{2+\cdots}} $$ or$\{1,1,2,3,4,5,\cdots,i,\cdots \} , i\in \mathbb{N}$ the simple continued fraction algebraic or transcendental? Any reference is appreciated EDIT and ...
0
votes
1answer
24 views

Any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Are there any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?
1
vote
1answer
52 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
1
vote
1answer
65 views

The relationship between each harmonic numbers

In Knuth's "Concrete Mathematics" in chapter about numbers below equality is given $$H_n = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + \frac{\epsilon_n}{120n^4} $$ where $0 < \epsilon_n < ...
3
votes
1answer
110 views

History of $p$-adic numbers

I'm interested in learning about the historical motivation and development of $p$-adic numbers. I haven't been able to find any books on the topic. I'd appreciate any references, including to more ...
2
votes
0answers
44 views

A better Reference than Andre Weil's Basic Number Theory

I want to get a feel for Adeles. I have been suggested to read the first 4 chapters of Andre Weil's Basic Number Theory. I am very confused by the writing style and conventions (like a field need not ...
0
votes
1answer
39 views

References for Legendre's prime-counting function

This question is about Legendre's prime-counting function, the one that can be used to calculate the exact amount of prime numbers that are less than or equal to a given number (as long as the number ...
0
votes
0answers
19 views

Reference request for papers proving the existence of a special function satisfying the following conditions

Previously in this site it has been proved that there exists at least one prime between $c_n$ and $n$ where $c_n$ denotes the $n$-th composite (see the question Prove that there exists an $m$ such ...
5
votes
1answer
93 views

Gentle introduction to algebraic number theory

For context, I am a undergraduate majoring in math. I've taken two semesters of algebra (though I am still a bit shaky on Galois theory). I just finshed a course in elementary number theory which used ...
4
votes
1answer
70 views

Undergraduate Introduction to Modular Forms

What are the best introductory texts (or lecture notes) on modular forms aimed at an advanced undergraduate audience (for a student with a course in complex analysis and two courses in algebra and ...
5
votes
1answer
107 views

Prove of trancendence of $\ln(2)$.

Where can I find some proofs for another transcendental numbers, like Hermite/Lindemann theorem proofs for $e/\pi$? For instance, prove that $\zeta(3)/\ln(2)$ is a transcendental number.
1
vote
0answers
118 views

Integral points on varieties and solutions to Diophantine equations

I am looking for a book (or article, or notes...) explaining details about the link between integral points on varieties defined as complement of certain divisors and integral solutions to the ...
2
votes
2answers
100 views

Second Course in Number Theory - Self Study

I just finished a first course in number theory using Dudley's Elementary Number Theory. This was by far my favorite math course and I want to learn more number theory this summer. As far as ...
3
votes
1answer
56 views

A property for perfect numbers

If $n=\prod\limits_{i=1}^k p_i^{\alpha_i}$, is a perfect number, with $p_i$ distinct primes, $\alpha_i\geq1$, then for each $i$ there is a $j \in \{1,\ldots,k\}$ such that $p_i \mid ...
2
votes
0answers
65 views

Consequence of the Riemann Hypothesis

So I watched this video: http://m.youtube.com/watch?v=rGo2hsoJSbo And it included the fact that a consequence of RH is that there will always be a prime number between consecutive cubic numbers. I ...
3
votes
2answers
63 views

The torsion of an elliptic curve over a finite field

There is a result for $p$ prime, $E$ an elliptic curve over $\mathbb F_p$, then $E(\overline{\mathbb{F}_p})[m]\cong (\mathbb{Z}/m\mathbb{Z})^2$ for $m \nmid p$. The book on cryptography I am using ...
6
votes
1answer
267 views

What are the active branches of number theory?

Context: I am a junior math major and am hoping to go to grad school after next year for a PhD. I have completed most of the standard undergraduate courses and have been consistently most interested ...
2
votes
2answers
150 views

Logic and number theory books

I've recently decided to start preparing for uni, so I figured I need to learn logic and some number theory. I picked up Burton's Elementary Number Theory and wasn't quite comfortable with it, seemed ...
2
votes
1answer
123 views

Twin Prime conjecture current status

Can someone help me with a link to read about the status of the Twin Prime conjecture. I have browse on the internet and have read some articles but still I have no clue of the updated status of Twin ...
4
votes
1answer
53 views

A (possibly) easier version of Bertrand's Postulate

While attending a math puzzle contest, my friend (a math student) asked me to prove that $$\sum_{k=1}^n \frac{1}{k} \notin \mathbb{Z} \quad \forall n \geq 2$$ Being the first time seeing this ...
18
votes
2answers
598 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
1
vote
0answers
66 views

The smallest class of numbers closed under addition, multiplication, and exponentiation

Let $\def\A{\mathfrak A}\A$ be the smallest subset of $\Bbb C$ that contains the algebraic numbers and also all numbers of the form $$\sum \alpha_i^{\beta_i}$$ where the $\alpha_i, \beta_i$ are ...
1
vote
2answers
148 views

Recommendation for Number Theory Textbook

. Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied ...
0
votes
0answers
43 views

$\lim _{x \to \infty} \sum_{p\leq x, p \equiv 1(mod k)} \frac {log(p)}{p}= \infty $

Where can I find a proof of the following equation? $\lim _{x \to \infty}\sum_{p\leq x,p \equiv 1(mod k)} \frac {log(p)}{p}= \infty $ where p is prime. The proof should be as elementary as ...
6
votes
1answer
72 views

Gowers' proof of Szemerdi's theorem

Are there any good books or other resources (expository notes) which explains Gowers' proof of Szemerdi's theorem in detail?
0
votes
1answer
44 views

A question about an asymptotic formula

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. ...
2
votes
1answer
44 views

Primes between consecutive cubes

I am looking at Dudley's proof of the existence of Mill's constant. It starts out as follows The proof depends on the following theorem: there is an integer $A$ such that if $n>A$, then there ...